Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.
Find your way through the grid starting at 2 and following these
operations. What number do you end on?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
A dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?
Hover your mouse over the counters to see which ones will be
removed. Click to remover them. The winner is the last one to
remove a counter. How you can make sure you win?
A tetromino is made up of four squares joined edge to edge. Can
this tetromino, together with 15 copies of itself, be used to cover
an eight by eight chessboard?
In how many ways can you fit two of these yellow triangles
together? Can you predict the number of ways two blue triangles can
be fitted together?
Can you shunt the trucks so that the Cattle truck and the Sheep
truck change places and the Engine is back on the main line?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Can you cover the camel with these pieces?
What happens when you try and fit the triomino pieces into these
Can you make a 3x3 cube with these shapes made from small cubes?
Here you see the front and back views of a dodecahedron. Each
vertex has been numbered so that the numbers around each pentagonal
face add up to 65. Can you find all the missing numbers?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
10 space travellers are waiting to board their spaceships. There
are two rows of seats in the waiting room. Using the rules, where
are they all sitting? Can you find all the possible ways?
Take a rectangle of paper and fold it in half, and half again, to
make four smaller rectangles. How many different ways can you fold
What is the best way to shunt these carriages so that each train
can continue its journey?
In a square in which the houses are evenly spaced, numbers 3 and 10
are opposite each other. What is the smallest and what is the
largest possible number of houses in the square?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
What is the least number of moves you can take to rearrange the
bears so that no bear is next to a bear of the same colour?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
A magician took a suit of thirteen cards and held them in his hand
face down. Every card he revealed had the same value as the one he
had just finished spelling. How did this work?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
This article for teachers describes how modelling number properties
involving multiplication using an array of objects not only allows
children to represent their thinking with concrete materials,. . . .
Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
How many different triangles can you make on a circular pegboard that has nine pegs?
A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.
Move just three of the circles so that the triangle faces in the
Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.
Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?
We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?
Imagine a wheel with different markings painted on it at regular
intervals. Can you predict the colour of the 18th mark? The 100th
A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?
Investigate the number of paths you can take from one vertex to
another in these 3D shapes. Is it possible to take an odd number
and an even number of paths to the same vertex?
A game for two players. You'll need some counters.
Exchange the positions of the two sets of counters in the least possible number of moves
If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?
An extension of noughts and crosses in which the grid is enlarged
and the length of the winning line can to altered to 3, 4 or 5.